7 Cauchy’s theorem

Definition 7.1 Let $D\subset\C$ and $\ga_0, \ga_1\colon[a,b]\to D$ curves in $D.$

A homotopy in $D$ between paths $\ga_0, \ga_1$ is a continuous map \[\Ga\colon[0,1]\t[a,b]\longra D, (s,t)\longmapsto\Ga_s(t),\] such that $\Ga_0(t)=\ga_0(t)$ and $\Ga_1(t)=\ga_1(t)$ for all $t\in[a,b].$ Then $\ga_0, \ga_1$ are called (freely) homotopic paths in $D.$

If, additionally, $\Ga_s(a)=p$ and $\Ga_s(b)=q$ are constant in $s\in[0,1],$ we call $\Ga$ a path homotopy in $D$ and $\ga_0, \ga_1$ path-homotopic in $D$.
If $\ga_0, \ga_1$ are loops, a homotopy of loops in $D$ is a homotopy $\Ga$ in $D$ with the additional property that $\Ga_s$ is a loop for each $s\in[0,1].$ Then $\ga_0, \ga_1$ are called (freely) homotopic loops in $D.$

A loop is null-homotopic in $D$ if there is a homotopy of loops in $D$ to the constant loop.

A path homotopy is a video of paths with fixed endpoints.

Remark 7.1. In the following we will also suppose that $\Ga$ is piecewise C¹, meaning there exist subdivisions \[0=s_0<s_1<\cdots<s_m=1,\qquad a=t_0<t_1<\cdots<t_n=b\]

such that each restriction $\Ga|_{[s_{j-1},s_j]\t[t_{k-1},t_k]}$ is continuously differentiable.

Example 7.1

A set $D$ is star-shaped if there exists a focal point $z_0\in D$ such that for each $z\in D$ the straight line segment $tz+(1-t)z_0,$ $t\in[0,1],$ is contained in $D.$ Disks, the complex plane, and rectangles are star-shaped. Every loop $\ga$ in a star-shaped domain is null-homotopic. If the focal point agrees with the base of the loop, the path homotopy is

\[\Ga_s(t) = (1-t)\ga(t) + tz_0. \tag{7.1}\]

In general, the path homotopy is more complicated to write down. We omit it, since we will not need this fact below.

Theorem 7.1 (Cauchy’s theorem) Let $f\colon U\to \C$ be a holomorphic function on an open set. Let $\ga_0, \ga_1$ be piecewise C¹ curves in $U$ that are path-homotopic in $U.$ Then

\[\int_{\ga_0}f(z)dz = \int_{\ga_1}f(z)dz.\]

Proof.

Pick a path homotopy $\Ga\colon[0,1]\t[a,b]\to U.$ By reparameterizing we may assume $[a,b]=[0,1].$ Write $R^{(0)}=[0,1]\t[0,1]$ for the domain of $\Ga$ and let $\6 R^{(0)}$ be its piecewise linear boundary path, with the obvious counterclockwise parameterization (see Figure 7.2).

Figure 7.2: Path homotopy when $\ga_0,$ $\ga_1$ are loops.

The boundary path has four segments and the path integral along each constant path vanishes. Using Proposition 6.2(b),(c), we then find that \[\int_{\ga_0} f(z)dz-\int_{\ga_1} f(z)dz=\int_{\Ga(\6 R^{(0)})} f(z)dz.\]

Subdivide $R^{(0)}$ into four congruent rectangles $R^{(0)}_i$ as in Figure 7.3. Using the fact that the path integrals in opposite directions cancel, we find \[\left|\int_{\Ga(\6 R^{(0)})} f(z)dz\right| = \left|\sum_{i=1}^4 \int_{\Ga(\6 R^{(0)}_i)} f(z)dz\right|\leqslant \sum_{i=1}^4 \left|\int_{\Ga(\6 R^{(0)}_i)} f(z)dz\right|.\]

Figure 7.3: Subdivision and cancellation of opposite paths

Let $R^{(1)}$ be the rectangle $R_i^{(0)}$ for which $\left|\int_{\Ga(\6 R^{(0)}_i)} f(z)dz\right|$ is maximal. Then \[\left|\int_{\Ga(\6 R^{(0)})} f(z)dz\right|\leqslant4\left|\int_{\Ga(\6 R^{(1)})} f(z)dz\right|.\]

Now repeat this process with $R^{(1)}$ to obtain $R^{(2)}$ and so forth. This yields a sequence of rectangles $R^{(n)}$ as in Figure 7.4 with sides of length $2^{-n}$ and \[\left|\int_{\Ga(\6 R^{(0)})} f(z)dz\right|\leqslant4^n\left|\int_{\Ga(\6 R^{(n)})} f(z)dz\right|. \tag{7.2}\]

Convergent sequence of rectangles $\bigcap R^{(n)}={(s_0,t_0)}$ — Figure 7.4: Convergent sequence of rectangles $\bigcap R^{(n)}=\{(s_0,t_0)\}$

As the side lengths tend to zero, the midpoints $(s^{(n)},t^{(n)})$ of the rectangles $R^{(n)}$ are a Cauchy sequence, so converge to some limit $(s_0,t_0).$ Let $z_0=\Ga_{s_0}(t_0).$ Since $f$ is complex differentiable at $z_0,$ we may write \[f(z)=f(z_0) + (z-z_0)f'(z_0) + (z-z_0)\rho(z),\qquad\lim_{z\to z_0}\rho(z)=0. \tag{7.3}\] By Example 6.4,\[\int_{\Ga(\6 R^{(n)})}\left(f(z_0) + (z-z_0)f'(z_0)\right)dz=0. \tag{7.4}\]

As $\Ga$ is piecewise C¹, its derivative is bounded in norm by some $C>0.$ Let $\ep>0.$ Pick $n_0\in\N$ such that for all $n\geqslant n_0$ and $z\in R^{(n)}$ we have $|\rho(z)|<\ep.$ As the side lengths of $R^{(n)}$ are $2^{-n}$ and $z_0\in R^{(n)},$ we have $|z-z_0|\leqslant\sqrt{2}2^{-n}$ for all $z\in\6 R^{(n)}.$ Combining this with Equation 7.3 and Equation 7.4, we can estimate

\[\begin{align*} \left|\int_{\Ga(\6 R^{(n)})} f(z)dz\right|&= \left|\int_{\Ga(\6 R^{(n)})}(z-z_0)\rho(z)dz\right|\\ &\leqslant L(\Ga(\6 R^{(n)}))\sqrt{2}2^{-n}\ep = 4C\cdot 2^{-n}\sqrt{2}2^{-n}\ep \end{align*}\]

The last inequality follows by Equation 6.7.

Hence \[\left|\int_\ga f(z)dz\right|=\left|\int_{\Ga(\6 R^{(0)})} f(z)dz\right|\overset{}{\leqslant}4C\sqrt{2}\ep.\] The final inequality follows from Equation 7.2.

As $\ep>0$ is arbitrary, the left hand side must be zero.

Theorem 7.2 Let $\ga_0, \ga_1$ be piecewise C¹ loops in $U$ that are freely homotopic in $U.$ Suppose $f\colon U\to\C$ is a holomorphic function. Then

\[\int_{\ga_0}f(z)dz = \int_{\ga_1}f(z)dz.\]In particular, if $\ga$ is a loop that is homotopic in $U$ to the constant loop, then\[\int_\ga f(z)dz=0. \tag{7.5}\]

Proof.

Let $\Ga$ be the homotopy between $\ga_0$ and $\ga_1.$ Let $\eta(s)=\Ga_s(0).$ Figure 7.5 describes the construction of a path homotopy between $\eta\ast\ga_0\ast(-\eta)$ and $\ga_1.$

Figure 7.5: Construction of path homotopy

Hence Theorem 7.1 implies our claim, using the fact that the path integrals over $\eta$ and $-\eta$ cancel each other.

Questions for further discussion

How should Figure 7.5 be interpreted? Can you give a formula for the path homotopy (consider cases)?

Give a counterexample to Cauchy’s theorem when (i) $\ga$ is not null-homotopic (ii) $f(z)$ is not holomorphic
Does Cauchy’s theorem hold for $f(z)=\ol{z}$?

7.1 Exercises

Exercise 7.1

Sketch the curves \[\begin{align*} \ga_0\colon[-1,1]&\longra\C,&\ga_0(t)&=t,\\ \ga_1\colon[-1,1]&\longra\C,&\ga_1(t)&=e^{i\pi\frac{1-t}{2}} \end{align*}\] and show that they are path-homotopic.
1. Let $0<r_0<r_1$ and $z_0\in\C.$ Prove that the loops $\partial D_{r_0}(z_0)$ and $\partial D_{r_1}(z_0)$ are freely homotopic in the closed annulus $\ol A_{r_0,r_1}(z_0).$

Exercise 7.2

Let $\al+i\be\in\C.$ Determine \[\int_a^b e^{(\al+i\be)t} dt\] to compute \[\int_a^b e^{\al t}\cos(\be t)dt.\]

Exercise 7.3

Let $\C^-=\C\setminus(-\iy,0]$ be the slit plane.

Show that any two points in $\C^-$ may be connected by a path in $\C^-.$ Hence $\C^-$ is path-connected.
Show that every closed curve $\ga\colon[a,b]\to\C^-$ is null-homotopic in $\C^-$ by finding a homotopy \[\Ga\colon[0,1]\t[a,b]\longra\C^-, (s,t)\longmapsto\Ga_s(t)\] satisfying $\Ga_0(t)=\ga(t)$ and $\Ga_1(t)=1$ for all $t\in[a,b].$ Hence $\C^-$ is simply-connected.
Prove the analogues of a. and b. for a disk $D_r(z_0).$
Use Cauchy’s Theorem to prove that the punctured plane $\C^\t=\C\setminus\{0\}$ is not simply-connected. That is, there exists a closed curve in $\C^\t$ that is not null-homotopic in $\C^\t.$

Exercise 7.4

Let $0<b<1.$

Using the geometric series, find the power series expansion \[\frac{1}{z-1/b}=\sum_{n=0}^\iy a_n(z-b)^n\] with center $z_0=b$ and determine the radius of convergence $\rho.$
Use a. to show that for all $0<r<\rho$ \[\int_{\partial D_r(b)}\frac{dz}{(z-b)(z-1/b)}=\frac{2\pi i}{b-1/b}.\]
Use c. to compute \[\int_0^{2\pi}\frac{dt}{1-2b\cos(t)+b^2}.\]

Exercise 7.5

Let $\ga\colon[0,1]\to D$ be a curve in $D\subset\C$ and let $-\ga\colon[0,1]\to D,$ $(-\ga)(t)=\ga(1-t)$ be the opposite curve. Prove that $\ga\ast(-\ga)$ is path-homotopic to a constant loop.